Group Theoretical Methods and Their Applications [Hardcover]

1 Preliminaries.- 1.1 The Concept of Groups.- 1.1.1 Transformation Groups.- 1.2 Price Index in Economics.- 1.3 The Realization of Groups.- 1.4 Representation of Groups.- 1.5 Equivalence of Representations.- 1.6 Reducibility of Representations.- 1.7 Complete Reducibility.- 1.8 Basic Conclusions.- 1.9 Representations of Special Finite Groups.- 1.9.1 The Cyclic Group Cg.- 1.9.2 The Dihedral Group Dh.- 1.10 Kronecker Products.- 1.11 Unitary Representations.- 1.11.1 Unitary Representations and Unitary Matrices.- Problems.- 2 Linear Operators with Symmetries.- 2.1 Schurs Lemma.- 2.2 Symmetry of a Matrix.- 2.2.1 Representations of Abelian Groups.- 2.3 The Fundamental Theorem.- Problems.- 3 Symmetry Adapted Basis Functions.- 3.1 Illustration by Dihedral Groups.- 3.1.1 The Representation ?per.- 3.1.2 The Notion of the Orbit.- 3.1.3 Instructions on How to Use Table (3.7).- 3.1.4 Real Orthonormal Symmetry Adapted Basis.- 3.2 Application in Quantum Physics.- 3.3 Application to Finite Element Method.- 3.3.1 Discretization and Symmetry of the Problem.- 3.3.2 Elliptic Boundary Value Problem.- 3.3.3 Heat conduction.- 3.4 Perturbed Problems with Symmetry.- 3.5 Fast Fourier Transform on Finite Groups.- 3.5.1 Definitions and Properties.- 3.5.2 Direct and Fast Algorithm.- 3.5.3 The Classical Fast Fourier Transform (FFT).- 3.5.4 Applications and Remarks.- 4 Continuous Groups And Representations.- 4.1 Continuous Matrix Groups.- 4.1.1 Comments on U(n).- 4.1.2 Comments on SU(n).- 4.1.3 Connectedness of Continuous Groups.- 4.2 Relationship Between Some Groups.- 4.2.1 Relationship between SL(2, C) and the Lorentz group.- 4.2.2 Relationship Between SU(2) and SO(3).- 4.3 Constructing Representations.- 4.3.1 Irreducible Representations of SU(2).- 4.3.2 Irreducible Representations of SO(3).- 4.3.3 Complete Reduction of Representations of SU(2) and SO(3).- 4.4 Clebsch-Gordan Coefficients.- 4.4.1 Spherical Functions.- 4.4.2 The Kronecker Product ?l??m.- 4.4.3 Spherical Functions and Laplace OperalCÏ